Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Proposition 2.1.6.

  1. is normal subgroup of
  2. is open and closed relative to
  3. An element belongs to if and only if For some natural number and some self-adjoint elements
  4. Note first that is closed under multiplication. (Why? See the comment underneath Definition 2.1.2.)
    We must also show that if belongs to then so do and for every . (Normality)
    Let be a continuous path in from to .
    Then and are continuous paths in from to and to respectively. Not exactly sure how continuity is immediate, perhaps since each point in the path is a unitary, we can continuously take inverses and also conjugate by unitaries
    This proves 1.

Let be the set of elements in of form Where and are all self-adjoint elements in .
It follows from 1. and Lemma 2.1.3 (1), that . Easily
Since for every self-adjoint , we see that is a Group.

Take in and with . Then and by 2.1.3 Lemma (3) and its proof for some self-adjoint element . Hence This shows that is open relative to .
The complement is a disjoint union of cosets of the form with . Each of these cosets are homeomorphic to and therefore open relative to , consequently is closed in .
In conclusion, is a non-empty subse of , is closed and open in , and is connected. This implies that .
Therefore 2. and 3. hold.

Table Of Contents
Proposition 2.1.6.